What's the information content of chaotic systems, say, the (classical) 3-body problem?

On one hand, it seems like with initial position and velocities and Newton's laws, the system is fully defined, and so only a very small amount of information is needed to fully describe the evolution of the system.

On the other hand, there is no closed form solution, and so to generally obtain the configuration at some later point in time requires you to compute the configuration of all the inbetween times; there is no shortcut to instantly get the configuration at any time.

Perhaps our perception of time exists because the universe has to "crunch the numbers" so to speak to fully describe itself.